Integralen av $$$\frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}}$$$

Bestäm $$$\int \frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx$$$.

Skriv om integranden:

$${\color{red}{\int{\frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} d x}}} = {\color{red}{\int{2 \cos^{2}{\left(x \right)} d x}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=2$$$ och $$$f{\left(x \right)} = \cos^{2}{\left(x \right)}$$$:

$${\color{red}{\int{2 \cos^{2}{\left(x \right)} d x}}} = {\color{red}{\left(2 \int{\cos^{2}{\left(x \right)} d x}\right)}}$$

Använd potensreduceringsformeln $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$ med $$$\alpha=x$$$:

$$2 {\color{red}{\int{\cos^{2}{\left(x \right)} d x}}} = 2 {\color{red}{\int{\left(\frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}\right)d x}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(x \right)} = \cos{\left(2 x \right)} + 1$$$:

$$2 {\color{red}{\int{\left(\frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}\right)d x}}} = 2 {\color{red}{\left(\frac{\int{\left(\cos{\left(2 x \right)} + 1\right)d x}}{2}\right)}}$$

Integrera termvis:

$${\color{red}{\int{\left(\cos{\left(2 x \right)} + 1\right)d x}}} = {\color{red}{\left(\int{1 d x} + \int{\cos{\left(2 x \right)} d x}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dx = c x$$$ med $$$c=1$$$:

$$\int{\cos{\left(2 x \right)} d x} + {\color{red}{\int{1 d x}}} = \int{\cos{\left(2 x \right)} d x} + {\color{red}{x}}$$

Låt $$$u=2 x$$$ vara.

Då $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (stegen kan ses »), och vi har att $$$dx = \frac{du}{2}$$$.

Alltså,

$$x + {\color{red}{\int{\cos{\left(2 x \right)} d x}}} = x + {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$x + {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}} = x + {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}$$

Integralen av cosinus är $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$x + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{2} = x + \frac{{\color{red}{\sin{\left(u \right)}}}}{2}$$

Kom ihåg att $$$u=2 x$$$:

$$x + \frac{\sin{\left({\color{red}{u}} \right)}}{2} = x + \frac{\sin{\left({\color{red}{\left(2 x\right)}} \right)}}{2}$$

Alltså,

$$\int{\frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} d x} = x + \frac{\sin{\left(2 x \right)}}{2}$$

Lägg till integrationskonstanten:

$$\int{\frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} d x} = x + \frac{\sin{\left(2 x \right)}}{2}+C$$

$$$\int \frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx = \left(x + \frac{\sin{\left(2 x \right)}}{2}\right) + C$$$A